Import Question JSON

Current Question (ID: 10134)

Question:

$\text{The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring will be:}$

Options:

1. $2 : 1$
2. $\sqrt{5} : \sqrt{6}$
3. $2 : 3$
4. $1 : \sqrt{2}$

Solution:

\text{For a circular disc: } I_{disk} = (5/4)MR^2 = MK_{disk}^2 \text{For a circular ring: } I_{ring} = (3/2)MR^2 = MK_{ring}^2 \text{Using the parallel axis theorem for tangential axis:} \text{For disc: } I_{tangential} = (1/4)MR^2 + MR^2 = (5/4)MR^2 \text{For ring: } I_{tangential} = (1/2)MR^2 + MR^2 = (3/2)MR^2 \text{Therefore: } K_{disk}/K_{ring} = \sqrt{5/6} \text{The ratio is } \sqrt{5} : \sqrt{6}

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}